Alessandro Terracini's obituary of Guido Fubini

During Alessandro Terracini's years in Argentina he wrote an obituary of Guido Fubini in Spanish. It was published as 'Guido Fubini (1879-1943)', Revisita de la Unión Matemática Argentina 10 (1) (1944), 27-30. We give below an English translation of the obituary.

Guido Fubini (1879-1943), by Alejandro Terracini.

On 6 June last year (1943), Guido Fubini passed away in New York. He was born in Venice (Italy) on 19 January 1879. He studied in Pisa, where he was educated at the Scuola Normale Superiore taught mathematics by Ulisse Dini and particularly Luigi Bianchi. Bianchi's influence was felt on the direction of Fubini's scientific thought, even many years after he left the classrooms of Pisa, which confirms how important it can be for an original and independent spirit like Fubini to study with a great teacher. Shortly after graduating, Fubini began teaching at the universities of Catania and Genoa, until, when a new chair of Mathematical Analysis was established at the School of Engineering in Turin (different from the one Peano taught and continued to teach at the University of the same city), Fubini was called to take charge of it. He held this position until 1938, while teaching Advanced Analysis at the University. After being removed from his chair due to anti-Semitic laws, he moved to the United States, where he worked at the Institute for Advanced Study in Princeton from 1939 to 1941. In the spring of 1941 he also taught a course on External Ballistics at New York University. His gratitude towards his host country and his enthusiasm for the Western Hemisphere seemed to breathe life into him; however, his health had been irremediably broken by events, and in the first months of 1942 he left Princeton to settle in New York, where, despite the heart disease that plagued him, he continued to devote himself at home to his favourite studies until his death.

When I was preparing to briefly describe Fubini's scientific personality for the readers of the Revista de la Unión Matemática Argentina, in accordance with the honourable invitation of Professor Rey Pastor, I found myself faced with a rather difficult problem, due to the need to break down a single whole into elements. First of all, Fubini's person, full of enthusiasm and brimming with vivacity, can hardly be separated from his works with the brilliance of his style, with his crystal-clear way of posing problems and placing them in the framework of problems already known, with his way of tackling them without fear of difficulties. It almost seems that his figure is always in motion, that the rapid accent of his strong voice full of expressive inflections emerges from his pages. Fubini gives us confirmation that mathematics is not a frigid and impersonal thing. On the other hand, his thought is largely unitary, even when he seems to take an interest in different topics.

Whatever the case, a great many of the works from the first decade of his production revolve around the theory of groups: at first, he dealt particularly with continuous groups (e.g. spaces admitting a continuous group of movements, or a conformal group; groups of geodesic transformations, etc.); but soon Fubini's attention was directed rather to infinite discontinuous groups and automorphic functions. His book Introduzione alla teoría dei gruppi discontinui e delle funzioni automorfe (Pisa, 1908) closes precisely this period. Of course, with these investigations are intertwined other investigations of a different direction, e.g. on Hermitian forms and the metrics they define.

Another motive around which a good part of Fubini's activity has developed is integrated with the Dirichlet problem and other surrounding problems, particularly in relation to variational methods. Among his works relevant to this age of older ideas, most of which date back to distant years (it is worth recalling that at the beginning of the century Hilbert had revived, by founding it on new bases, that powerful Dirichlet principle which had been neglected after Weierstrass's devastating criticism), we recall in particular his contribution The Principle of the Minimum published in 1935 in the Revista Matemática Hispano-Americana (which had already published other works by Fubini, e.g. on differential geometry in 1921, and in 1929 the obituary of Bianchi; also in 1935 it published Fubini's On the Bending of a Beam of Small Curvature).

To this we must add various works on different topics of analysis (differential equations, double and multiple integrals, integral equations, series of functions, etc. etc.). At certain moments of his life, Fubini was attracted by other problems, such as those of exterior ballistics, and the theory of the beam.

However, there is another branch that represents almost thirty years of Fubini's mathematical thought (certainly without exhausting it), and that is differential projective geometry. In 1914 his first work Definizione proiettivo-differenziale di una superficie appeared, published in the Atti dell' Accademia di Torino, and already in 1931 a list of Fubini's works on this topic included 46 items (according to a provisional list, 8 works from after that year must be added). Not only that, but I dare say one thing which Fubini might not like very much, and that is that if there is one branch of mathematics in which Fubini's name will be remembered for a long time, it is precisely this one. We have often talked with Fubini about which results of such and such a mathematician will remain as an effective and definitive achievements in science, and I still seem to see him pessimistically and negatively shaking his head and affirming that almost everything would perish very soon. I did not think then that I would be called upon so soon to attempt a prediction in this respect, and I am very sorry that he would probably not like my judgment. He was born an analyst and lived as an analyst, and always felt deeply an analyst. However, his previous work in the classical direction of differential geometry (which had remained in him as one of the traits inherited from the Bianchi school) and perhaps his spirit of scientific adventure led him to systematically face and construct differential projective geometry, basing it on the theory of differential forms, analogously to what Gauss and his followers did for the metric theory of surfaces. Thus Fubini found himself acting as one of the founders of that branch (another is Wilczynski, who however resorted to different analytical methods), and after having created the tools he knew how to use them with the enthusiasm of a pioneer and the skill of a master. What can be said is that Fubini always seemed to me to be an analyst even in differential geometry, where the analytical spirit has very often guided him to create the most hidden geometric concepts. The theory that he was constructing is systematically exposed in the two works of the collection Geometry proiettiva differenziale (Bologna, 1926-27) and Introduction à la géométrie projective différentielle des surfaces (Paris, 1931), both written in collaboration with E Cech.

Fubini had first-rate teaching qualities: I believe that it is rare to find the qualities of researcher and lecturer combined in the same person as it was with Fubini. His students at the School of Engineering in Turin idolised him, and with good reason, since several thousand engineers who have graduated from that school owed their solid mathematical training to him. The vivacity of his writings was multiplied even more in his classes, where all the concepts came out with a highlights that gave them an unforgettable clarity and a drive that imprinted them definitively on the audience. His Lezioni di analisi matematica (STEN, Turin), of which several editions have followed since 1912, reproduce his courses given to engineering students, and remain a model of simplicity and clarity. The "Esercizi di analisi matematica" written in collaboration with G Vivanti (STEN, Turin, 2nd ed., 1930) were also very successful. Fubini also wrote a comprehensive exposition on Engineering Mathematics, in which he developed the essential parts of the knowledge useful to engineers: it is hoped that the book will soon be published in Spanish, and the publication will not fail to be extremely beneficial for students and engineers.

Fubini has been a generous and good man: his material and moral support has never failed those who turned to him. He deeply felt friendship; and he will never be forgotten by the friends who for many years have discussed with him, day after day, problems of science and life, who have shared his efforts and concerns, who - even through the great distances that have separated them - have always remained spiritually close to him.

Universidad Nacional de Tucumán, April 1944

Last Updated March 2025